Definition 17.10.1 (Cyclotomic extension). The th cyclotomic extension of is where is a primitive th root of unity. The minimal polynomial of over is the th cyclotomic polynomial , which has degree (Euler’s totient).
Proposition 17.10.1 (Galois group of cyclotomic extensions). The extension is Galois, and The isomorphism sends (defined by ) to .
Note. This is abelian, so every subextension of is Galois over . The Kronecker–Weber theorem states that every abelian extension of is contained in some cyclotomic extension.